Anthony Day was still walking around in a daze this week, unsure when he would be able to grasp the reality of the whole thing. Maybe it will come Saturday afternoon when he and his Yale hockey teammates are set to throw out the first pitch in Fenway. It’s entirely possible it will be never.

Day is an intelligent young man (see above: Yale), but you can’t blame him for having trouble getting his brain around this one. After all, what were the odds of winning a national championship in hockey when adding all the variables? You don’t need a Ph.D. in probability and statistics to see they hovered around zero.

Yale wasn’t even sure it would be invited to the 16-team NCAA Tournament after getting outscored, 8-0, in the final two games of the ECAC playoffs. It had never won a national title in hockey. Harvard was the last Ivy League school to win one, and that was 24 years ago. Before Harvard, it was Cornell in 1970 when Ken Dryden, now 65, was in goal.

“I always said that it would be unbelievable if we could win it,” Day said. “I’m always thinking of the best possible scenario, and my best friend always knocks on wood. It got to the point a couple weeks ago where I said, ‘You don’t have to knock on wood. It’s a reality now.’ To get into the tournament, on the bubble, it just rejuvenated us.”

The odds of Day playing big-time hockey were minuscule after he graduated from St. Francis High in Athol Springs. He grew up in Elma and was a good player but was basically ignored by Division I programs. Niagara and Canisius didn’t show interest for reasons he didn’t understand this week.

His best guess was that his size scared off teams even though many other 5-foot-9, 175-pound forwards have played D-I. He also was overlooked in the USHL draft but latched on with Sioux Falls after paying his way to an open league tryout. He had a good season and jumped at the opportunity to play for Yale because, well, it’s Yale.

“I think they were stacked against me,” he said of the odds. “Nobody showed me any interest.”

If you’ve learned anything about Day by now, it’s that he knows how to compete. It doesn’t matter if he’s in the corner of the offensive zone or the corner of the classroom. The size of his body says one thing, his heart another. It’s why he never quit even when common sense suggested his playing days were over.

Instead, he kept pushing.

His drive comes from his family. His mother is a teacher’s aide by day, a waitress by night at Danny’s South near Ralph Wilson Stadium. His father hasn’t worked since 1991, the year Anthony was born, after suffering a back injury that required four surgeries. Two older brothers, Steve Bennett and Greg Day, encouraged him to max out whatever potential he had for as long as he could.

Day’s boyhood fantasy wasn’t about winning the Stanley Cup. He loved the college game and wanted to play in the Frozen Four. When he was a freshman at Frannies, he watched Tim Kennedy, Chris Mueller and Mike Ratchuk win it all for Michigan State. He skated with them over the summer. They were evidence it was possible.

“The odds were crazy,” he said.

The odds seem longer considering he’s not a superstar. The sophomore is a role player who had one goal and four assists in 37 games this season. He’s thrilled with every game he plays at the Division I level because, for a while, he didn’t think he would ever play a single shift. It makes his story even more special.

You pull for kids like him, an underdog playing for an underdog. He’s not blessed with ridiculous skill that you see in the nation’s top players. Day had two assists in a 4-1 win over North Dakota, which had 14 NHL draft picks on the roster, in a second-round knockout. He helped Yale dump Minnesota in the first.

Yale still was a long shot after reaching the Frozen Four in Pittsburgh, but that means little in college hockey. Any team can win on a given night. It came down to a single shot when it beat UMass Lowell in overtime. But the Bulldogs’ amazing run was certain to end against top-ranked Quinnipiac in the championship game.

Right? Really, what were the chances of four straight upsets?

And there was the Buffalo thing, which followed around Day like a terrible rumor. He had lost four state championships, including three at St. Francis. He was born when the Bills were en route to losing four straight Super Bowls. Need we revisit the Sabres at this stage? Winning was a stranger on a street corner.

“It was absolutely foreign,” Day said. “I never won a state championship, which was something I always wanted. I was fitting right into Buffalo. Now, I’m an outcast with a national championship.”

Quinnipiac was vulnerable after falling two goals behind Canisius in the first round, a performance dismissed as an off night. Quinnipiac had beaten Yale three times by a combined 13-3, including a 3-0 win in the conference semifinals. Many were expecting a similar rout in the national championship game last Saturday. In fact, it was.

Yale 4, Quinnipiac 0.

With his family in the stands, with his teammates trying to comprehend reality, Day finally experienced the euphoria of winning. He’s still trying to make sense of it all. A kid who wasn’t recruited out of high school lands at an Ivy League school and wins a national title? It doesn’t make sense.

He had 165 text messages waiting for him after the game. His friends and family knew the odds. And they just watched Day overcome them.